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Quadrilateral ABCD is a rhombus and points C, D, and E are

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Quadrilateral ABCD is a rhombus and points C, D, and E are [#permalink]

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09 Mar 2010, 20:10
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Quadrilateral ABCD is a rhombus and points C, D, and E are on the same line. Is quadrilateral ABDE a rhombus?

(1) The measure of angle BCD is 60 degrees.
(2) AE is parallel to BD

OPEN DISCUSSION OF THIS QUESTION IS HERE: quadrilateral-abcd-is-a-rhombus-and-points-c-d-and-e-are-82126.html
[Reveal] Spoiler: OA

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Last edited by Bunuel on 13 Oct 2013, 16:03, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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09 Mar 2010, 20:36

Join BD with a broken line to indicate an imaginary figure ABDE. AD as diagonal, we need both the given statements to prove that t'gle ABD and t'gle ADE are both congruent. Hence, Angles B and E are equal and opposite, so are angles A and D, and sides AB = BD = DE = AE.

Hence, ABDE will be a rhombus, if the given stimulus and statements are true.
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13 Oct 2013, 15:59
vscid wrote:

What's the OA for this one please? Is there anyone else that might want to give it a shot?
I got (C) but really more because of intuition than because I can explain it clearly.

Many thanks
Will provide some Kudos to good answers
Cheers
J:)
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13 Oct 2013, 16:05
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jlgdr wrote:
vscid wrote:

What's the OA for this one please? Is there anyone else that might want to give it a shot?
I got (C) but really more because of intuition than because I can explain it clearly.

Many thanks
Will provide some Kudos to good answers
Cheers
J:)

Quadrilateral ABCD is a rhombus and points C, D, and E are on the same line. Is quadrilateral ABDE a rhombus?

Rhombus is a quadrilateral with all four sides equal in length. A rhombus is actually just a special type of parallelogram (just like square or rectangle).

So ABCD is a rhombus means AB=BC=CD=AD.
ABDE to be a rhombus it must be true that AB=BD=DE=AE.

(1) The measure of angle BCD is 60 degrees --> diagonal BD equals to the sides of rhombus, so BD=AB. Know nothing about DE or/and AE. Not sufficient.

(2) AE is parallel to BD --> ABDE is a parallelogram (as AE||BD and BA||DE), hence opposite sides are equal: BD=AE and AB=DE. But we don't know whether all sides are equal (AB=BD=DE=AE). Not sufficient.

(1)+(2) From (1): BD=AB and from (2) BD=AE and AB=DE --> AB=BD=DE=AE --> ABDE is a rhombus. Sufficient.

OPEN DISCUSSION OF THIS QUESTION IS HERE: quadrilateral-abcd-is-a-rhombus-and-points-c-d-and-e-are-82126.html
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Re: rhombus   [#permalink] 13 Oct 2013, 16:05
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